{
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "g(x) = cos(2cos(x^2)+(1/4)*(x-1)^2)+x/2+1\n",
    "low=0;high=4;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "make_uv (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function make_uv(r)\n",
    "    δ= (high-low)/(r-1)\n",
    "    u = low:δ:high\n",
    "    v = g.(u);\n",
    "    u,v\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "make_f (generic function with 1 method)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function make_f(u,v)\n",
    "    r = length(u)\n",
    "    σ(x) = max(x,0)\n",
    "    N0 = 1\n",
    "    N1 = r-1\n",
    "    N2 = 1;\n",
    "    W1 = ones(N1,1)\n",
    "    b1 = -[u[i] for i in 1:N1];\n",
    "    s(i) = (i == 0 ? 0 : (v[i+1]-v[i])/(u[i+1]-u[i])) \n",
    "    W2 = hcat([s(i) - s(i-1) for i in 1:N1]...)\n",
    "    b2 = v[1]\n",
    "    f(x) = (W2*σ.(W1*x + b1) .+ b2)[1,1]\n",
    "    return f\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(::var\"#f#7\"{Float64, Matrix{Float64}, Vector{Float64}, Matrix{Float64}, var\"#σ#3\"}) (generic function with 1 method)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f25 = make_f(make_uv(25)...)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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       "  1043.39,827.616 1044.68,817.939 1045.98,808.012 1047.27,797.854 1048.57,787.488 1049.86,776.939 1051.16,766.234 1052.45,755.401 1053.74,744.472 1055.04,733.479 \n",
       "  1056.33,722.457 1057.63,711.441 1058.92,700.469 1060.22,689.578 1061.51,678.806 1062.81,668.192 1064.1,657.774 1065.4,647.591 1066.69,637.68 1067.98,628.078 \n",
       "  1069.28,618.818 1070.57,609.936 1071.87,601.461 1073.16,593.423 1074.46,585.848 1075.75,578.76 1077.05,572.179 1078.34,566.123 1079.63,560.605 1080.93,555.636 \n",
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       "  1944.37,1219.2 1945.66,1218.21 1946.95,1217.22 1948.25,1216.24 1949.54,1215.25 1950.84,1214.27 1952.13,1213.29 1953.43,1212.31 1954.72,1211.33 1956.02,1210.35 \n",
       "  1957.31,1209.37 1958.6,1208.4 1959.9,1207.42 1961.19,1206.45 1962.49,1205.48 1963.78,1204.51 1965.08,1203.54 1966.37,1202.57 1967.67,1201.6 1968.96,1200.64 \n",
       "  1970.25,1199.67 1971.55,1198.71 1972.84,1197.74 1974.14,1196.78 1975.43,1195.82 1976.73,1194.86 1978.02,1193.9 1979.32,1192.94 1980.61,1191.98 1981.91,1191.03 \n",
       "  1983.2,1190.07 1984.49,1189.11 1985.79,1188.16 1987.08,1187.2 1988.38,1186.25 1989.67,1185.29 1990.97,1184.33 1992.26,1183.38 1993.56,1182.42 1994.85,1181.47 \n",
       "  1996.14,1180.51 1997.44,1179.55 1998.73,1178.6 2000.03,1177.64 2001.32,1176.68 2002.62,1175.72 2003.91,1174.76 2005.21,1173.8 2006.5,1172.83 2007.8,1171.87 \n",
       "  2009.09,1170.9 2010.38,1169.93 2011.68,1168.96 2012.97,1167.99 2014.27,1167.02 2015.56,1166.04 2016.86,1165.06 2018.15,1164.08 2019.45,1163.09 2020.74,1162.1 \n",
       "  2022.03,1161.11 2023.33,1160.12 2024.62,1159.12 2025.92,1158.12 2027.21,1157.11 2028.51,1156.1 2029.8,1155.09 2031.1,1154.07 2032.39,1153.05 2033.68,1152.02 \n",
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       "  2047.92,1140.33 2049.22,1139.22 2050.51,1138.11 2051.81,1136.99 2053.1,1135.86 2054.4,1134.72 2055.69,1133.58 2056.99,1132.42 2058.28,1131.26 2059.57,1130.08 \n",
       "  2060.87,1128.9 2062.16,1127.7 2063.46,1126.49 2064.75,1125.28 2066.05,1124.05 2067.34,1122.81 2068.64,1121.55 2069.93,1120.29 2071.23,1119.01 2072.52,1117.72 \n",
       "  2073.81,1116.42 2075.11,1115.1 2076.4,1113.77 2077.7,1112.43 2078.99,1111.07 2080.29,1109.69 2081.58,1108.3 2082.88,1106.9 2084.17,1105.47 2085.46,1104.04 \n",
       "  2086.76,1102.58 2088.05,1101.11 2089.35,1099.62 2090.64,1098.11 2091.94,1096.59 2093.23,1095.04 2094.53,1093.48 2095.82,1091.9 2097.11,1090.3 2098.41,1088.68 \n",
       "  2099.7,1087.03 2101,1085.37 2102.29,1083.68 2103.59,1081.98 2104.88,1080.25 2106.18,1078.5 2107.47,1076.72 2108.77,1074.93 2110.06,1073.1 2111.35,1071.26 \n",
       "  2112.65,1069.39 2113.94,1067.49 2115.24,1065.57 2116.53,1063.63 2117.83,1061.65 2119.12,1059.66 2120.42,1057.63 2121.71,1055.58 2123,1053.5 2124.3,1051.39 \n",
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      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots, Measures, LaTeXStrings\n",
    "xgrid=low:0.005:high\n",
    "\n",
    "u,v = make_uv(10)\n",
    "f25 = make_f(u,v)\n",
    "p1 = plot(xgrid,g.(xgrid),label=L\"f^*(x)\",lw=2)\n",
    "     plot!(xgrid,f25.(xgrid),label=\"f(x ; r = 10)\",lw=2)\n",
    "     scatter!(u,v,label = L\"(u_1,v_1),\\ldots,(u_{10},v_{10})\")\n",
    "\n",
    "u,v = make_uv(30)\n",
    "f50 = make_f(u,v)\n",
    "p2 = plot(xgrid,g.(xgrid),label=L\"f^*(x)\",lw=2)\n",
    "     plot!(xgrid,f50.(xgrid),label=\"f(x ; r = 30)\",lw=2)\n",
    "     scatter!(u,v,label = L\"(u_1,v_1),\\ldots,(u_{30},v_{30})\")\n",
    "\n",
    "plot(p1,p2,size=(800,400),\n",
    "    margin=10mm,\n",
    "    xlabel=\"Input x\",ylabel=\"Output\",\n",
    "    legend=:topleft,ylim=(0,5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "plot(p1,size=(400,400),\n",
    "    xlabel=\"Input x\",ylabel=\"Output\",\n",
    "    legend=:topleft,ylim=(0,5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "savefig(\"relu-approximation_a.pdf\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(p2,size=(400,400),\n",
    "    xlabel=\"Input x\",ylabel=\"Output\",\n",
    "    legend=:topleft,ylim=(0,5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "savefig(\"relu-approximation_b.pdf\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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